Intermolecular Energy and Virial Function (Van der Waals)

In this section, the virial and energy equation of Van der Waals interaction for different potential function are discussed in details.

VDW

This option calculates potential energy without any truncation.

Potential Calculation

Interactions between atoms can be modeled with an n−6 potential, a Mie potential in which the attractive exponent is fixed. The Mie potential can be viewed as a generalized version of the 12-6 Lennard-Jones potential,

\[E_{ij} = C_{n_{ij}} \epsilon_{ij} \bigg[\bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^{n_{ij}} - \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^6\bigg]\]

where \(r_{ij}\), \(\epsilon_{ij}\), and \(\sigma_{ij}\) are, respectively, the separation, well depth, and collision diameter for the pair of interaction sites \(i\) and \(j\). The constant \(C_n\) is a normalization factor such that the minimum of the potential remains at \(−\epsilon_{ij}\) for all \(n_{ij}\). In the 12-6 potential, \(C_n\) reduces to the familiar value of 4.

\[C_{n_{ij}} = \bigg(\frac{n_{ij}}{n_{ij} - 6} \bigg)\bigg(\frac{n_{ij}}{6} \bigg)^{6/(n_{ij} - 6)}\]
Virial Calculation

Virial is basically the negative derivative of energy with respect to distance, multiplied by distance.

\[W_{ij} = -\frac{dE_{ij}}{dr}\times \frac{\rightharpoonup{r_{ij}}}{r_{ij}}\]

Using n−6 LJ potential defined above:

\[W_{ij} = 6C_{n_{ij}} \epsilon_{ij} \bigg[\frac{n_{ij}}{6} \times \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^{n_{ij}} - \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^6\bigg]\times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}^2}\]

Note

This option only evaluates the energy up to specified Rcut distance. Tail correction to energy and pressure can be specified to account for infinite cutoff distance.

SHIFT

This option forces the potential energy to be zero at Rcut distance.

Potential Calculation

Interactions between atoms can be modeled with an n−6 potential,

\[E_{ij}(\texttt{shift}) = C_{n_{ij}} \epsilon_{ij} \bigg[\bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^{n_{ij}} - \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^6\bigg] - C_{n_{ij}} \epsilon_{ij} \bigg[\bigg(\frac{\sigma_{ij}}{r_{cut}}\bigg)^{n_{ij}} - \bigg(\frac{\sigma_{ij}}{r_{cut}}\bigg)^6\bigg]\]

where \(r_{ij}\), \(\epsilon_{ij}\), and \(\sigma_{ij}\) are, respectively, the separation, well depth, and collision diameter for the pair of interaction sites \(i\) and \(j\). The constant \(C_n\) is a normalization factor according to Eq. 3, such that the minimum of the potential remains at \(−\epsilon_{ij}\) for all \(n_{ij}\). In the 12-6 potential, \(C_n\) reduces to the familiar value of 4.

Virial Calculation

Virial is basically the negative derivative of energy with respect to distance, multiplied by distance, Eq. 4.

Using SHIFT potential function defined above:

\[W_{ij}(\texttt{shift}) = 6C_{n_{ij}} \epsilon_{ij} \bigg[\frac{n_{ij}}{6} \times \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^{n_{ij}} - \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^6\bigg]\times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}^2}\]
_images/VDW_SHIFT.png

Graph of Van der Waals potential with and without the application of the SHIFT function. With the SHIFT function active, the potential by force was reduced to 0.0 at the Rcut distance. With the SHIFT function, there is a discontinuity where the potential is truncated.

SWITCH

This option in CHARMM or EXOTIC force field smoothly forces the potential energy to be zero at Rcut distance and starts modifying the potential at Rswitch distance.

Potential Calculation

Interactions between atoms can be modeled with an n−6 potential,

\[E_{ij}(\texttt{switch}) = C_{n_{ij}} \epsilon_{ij} \bigg[\bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^{n_{ij}} - \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^6\bigg]\times F_E\]

where \(r_{ij}\), \(\epsilon_{ij}\), and \(\sigma_{ij}\) are, respectively, the separation, well depth, and collision diameter for the pair of interaction sites \(i\) and \(j\). The constant \(C_n\) is a normalization factor according to Eq. 3, such that the minimum of the potential remains at \(−\epsilon_{ij}\) for all \(n_{ij}\). In the 12-6 potential, \(C_n\) reduces to the familiar value of 4.

The factor \(F_E\) is defined as:

\[\begin{split}F_E = \begin{cases} 1 & r_{ij} \leq r_{switch} \\ \frac{\big({r_{cut}}^2 - {r_{ij}}^2 \big)^2 \times \big({r_{cut}}^2 - 3{r_{switch}}^2 + 2{r_{ij}}^2 \big)}{\big({r_{cut}}^2 - {r_{switch}}^2 \big)^3} & r_{switch} < r_{ij} < r_{cut} \\ 0 & r_{ij} \geq r_{cut} \end{cases}\end{split}\]
Virial Calculation

Virial is basically the negative derivative of energy with respect to distance, multiplied by distance, Eq. 4.

Using SWITCH potential function defined above:

\[W_{ij}(\texttt{switch}) = \Bigg[6 C_{n_{ij}} \epsilon_{ij} \bigg[\frac{n_{ij}}{6} \times \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^{n_{ij}} - \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^6\bigg]\times \frac{F_E}{{r_{ij}}^2} - C_{n_{ij}} \epsilon_{ij} \bigg[\bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^{n_{ij}} - \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^6\bigg] \times F_W \Bigg] \times \overrightarrow{r_{ij}}\]

The factor \(F_W\) is defined as:

\[\begin{split}F_W = \begin{cases} 1 & r_{ij} \leq r_{switch} \\ \frac{12\big({r_{cut}}^2 - {r_{ij}}^2 \big) \times \big({r_{switch}}^2 - {r_{ij}}^2 \big)}{\big({r_{cut}}^2 - {r_{switch}}^2 \big)^3} & r_{switch} < r_{ij} < r_{cut} \\ 0 & r_{ij} \geq r_{cut} \end{cases}\end{split}\]
_images/SWITCH.png

Graph of Van der Waals potential with and without the application of the SWITCH function. With the SWITCH function active, the potential is smoothly reduced to 0.0 at the Rcut distance.

SWITCH (MARTINI)

This option in MARTINI force field smoothly forces the potential energy to be zero at Rcut distance and starts modifying the potential at Rswitch distance.

Potential Calculation

Potential Calculation: Interactions between atoms can be modeled with an n−6 potential. In standard MARTINI, \(n\) is equal to 12,

\[E_{ij}(\texttt{switch}) = C_{n_{ij}}\epsilon_{ij} \Bigg[ {\sigma_{ij}}^{n} \bigg(\frac{1}{{r_{ij}}^{n}} + \varphi_{n} (r_{ij}) \bigg) - {\sigma_{ij}}^{6} \bigg(\frac{1}{{r_{ij}}^{6}} + \varphi_{6} (r_{ij}) \bigg) \Bigg]\]

where \(r_{ij}\), \(\epsilon_{ij}\), and \(\sigma_{ij}\) are, respectively, the separation, well depth, and collision diameter for the pair of interaction sites \(i\) and \(j\). The constant \(C_n\) is a normalization factor according to Eq. 3, such that the minimum of the potential remains at \(−\epsilon_{ij}\) for all \(n_{ij}\). In the 12-6 potential, \(C_n\) reduces to the familiar value of 4.

The factor \(\varphi_{\alpha}\) and constants are defined as:

\[\begin{split}\varphi_{\alpha}(r_{ij}) = \begin{cases} -C_{\alpha} & r_{ij} \leq r_{switch} \\ -\frac{A_{\alpha}}{3} (r_{ij} - r_{switch})^3 -\frac{B_{\alpha}}{4} (r_{ij} - r_{switch})^4 - C_{\alpha} & r_{switch} < r_{ij} < r_{cut} \\ 0 & r_{ij} \geq r_{cut} \end{cases}\end{split}\]
\[A_{\alpha} = \alpha \frac{(\alpha + 1) r_{switch} - (\alpha +4) r_{cut}} {{r_{cut}}^{(\alpha + 2)} {(r_{cut} - r_{switch})}^2}\]
\[B_{\alpha} = \alpha \frac{(\alpha + 1) r_{switch} - (\alpha +3) r_{cut}} {{r_{cut}}^{(\alpha + 2)} {(r_{cut} - r_{switch})}^3}\]
\[C_{\alpha} = \frac{1}{{r_{cut}}^{\alpha}} -\frac{A_{\alpha}}{3} (r_{cut} - r_{switch})^3 -\frac{B_{\alpha}}{4} (r_{cut} - r_{switch})^4\]
Virial Calculation

Virial is basically the negative derivative of energy with respect to distance, mul- tiplied by distance, Eq. 4.

Using the SWITCH potential function defined for MARTINI force field:

\[W_{ij}(\texttt{switch}) = C_{n_{ij}}\epsilon_{ij} \Bigg[ {\sigma_{ij}}^{n} \bigg(\frac{n}{{r_{ij}}^{(n+1)}} + d\varphi_{n} (r_{ij}) \bigg) - {\sigma_{ij}}^{6} \bigg(\frac{6}{{r_{ij}}^{(6+1)}} +d \varphi_{6} (r_{ij}) \bigg) \Bigg]\times \frac{\overrightarrow{r_{ij}}}{r_{ij}}\]

The constants defined in Eq. 14-16 and the factor \(d\varphi_{\alpha}\) defined as:

\[\begin{split}d\varphi_{\alpha}(r_{ij}) = \begin{cases} 0 & r_{ij} \leq r_{switch} \\ A_{\alpha} (r_{ij} - r_{switch})^2 + B_{\alpha} (r_{ij} - r_{switch})^3 & r_{switch} < r_{ij} < r_{cut} \\ 0 & r_{ij} \geq r_{cut} \end{cases}\end{split}\]
_images/MARTINI.png

Graph of Van der Waals potential with and without the application of the SWITCH function in MARTINI force field. With the SWITCH function active, the potential is smoothly reduced to 0.0 at the Rcut distance.